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360edo divides the octave into steps of 3 1/3 cents. 360 has many proper divisors: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180. 360 is the 13th [[superabundant EDO]].
{{Infobox ET}}
{{ED intro}}


360edo is used in the [[wikipedia:Eyeborg|eyeborg]], which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.
== Theory ==
360edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. It can also be used with 2.5.9.13 subgroup.  


== Theory ==
In the 5-limit, the [[patent val]] [[support]]s the [[misty]] temperament, and in the 7-limit 360edo supports the [[trimisty]] (name proposed by Eliora) 63 & 99 temperament with the comma basis {[[10976/10935]], 2097152/2083725}, which is similar to the misty temperament but has a period of 1/9- rather than 1/3-octave.
{{Primes in edo|360|columns=10}}
 
Its 23-limit patent val is <360 571 836 1011 1245 1332 1471 1529 1628|. This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421. Its 5-limit patent val [[support]]s misty temperament, and the 7-limit patent val supports the 63&99 temperament with wedgie <<9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&360 = <<11 76 51 95 50 -95|| and 12&360 = <<12 -48 -108 -104 -205 -116||; neither is very good though 41&360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&140 temperament, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit.  
360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the {{nowrap|140 & 220}} temperament with period 1\20. Aside from that, it provides the optimal patent val for the {{nowrap|41 & 360}} temperament with comma basis {10976/10935, 16384000000/16209796869}, on which it has lower badness than any other 7-limit temperament for which 360edo gives the optimal patent val. It also supports {{nowrap|12 & 360}} with the comma basis {[[390625/388962]], 67108864/66430125}. 
 
Aside from the patent val, there is a number of mappings to be considered. The 360d val, {{val|360 571 836 '''1010'''}}, tempers out 3136/3125, 5120/5103, and extends the misty temperament in to the 7-limit. It is also a tuning for the 12th-octave [[magnesium]] temperament.  


In the 360b val, 360edo's fifth is the same as 12edo. Coincidentally, the difference between a just fifth and a 12edo one is known as the grad, being a variant of translation of "degree", and 1/360th of a circle is a degree.
=== Odd harmonics ===
{{Harmonics in equal|360}}


==== Proposed notation ====
=== Subsets and supersets ===
Eliora proposes notating 360edo with calendar dates, Jan 1 being the tonic, Jan 2 being the next step, etc, and each month having even 30 days. The notation is convenient because 1 month in this scenario is equal to 1 semitone, and corresponds to [[12edo]].
360 is the 13th [[highly composite edo]], with many proper divisors: {{EDOs| 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 }}. One step of 360edo is known as '''the Dröbisch angle''', an [[interval size measure]] first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle".


== Table of intervals ==
== Table of intervals ==
{| class="wikitable"
[[Eliora]] proposes notating 360edo with calendar dates, Jan 1 being the tonic, Jan 2 being the next step, etc, and each month having even 30 days. The notation is convenient because 1 month in this scenario is equal to 1 semitone, and corresponds to [[12edo]].
|+Table of selected intervals
 
!Step
Any other notation system involving the number 360 can also be used.
!Name
 
!Calendar notation (if unison is Jan 1)
See: [[Table of 360edo intervals]]
!Ratio
 
== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
|0
! Periods<br />per 8ve
|Prime, unison
! Generator*
|January 1
! Cents*
|1/1
! Associated<br />ratio*
! Temperaments
|-
|-
|1
| 1
|Degree, grad, schisma
| 119\360
|January 2
| 396.67
|32805/32768
| 44/35
| [[Squarschmidt]]
|-
|-
|30
| 2
|Dodecaphonic semitone
| 53\360
|February 1
| 176.67
|89/84
| 448/405
| Quatracot
|-
|-
|36
| 3
|Septimal diatonic semitone, decioctave
| 149\360<br />(29\360)
|February 6
| 703.33<br />(303.33)
|[[15/14]]
| 4/3<br />(135/128)
| [[Misty]]
|-
|-
|60
| 4
|Dodecaphonic major second
| 23\360
|March 1
| 76.67
|
| 4302592/4100625
| [[Reenactment]]
|-
|-
|90
| 9
|Dodecaphonic minor third
| 149\360<br />(29\360)
|April 1
| 703.33<br />(36.67)
|
| 4/3<br />(135/128)
| [[Trimisty]]
|-
|-
|116
| 12
|Classical major third
| 73\360<br />(13\360)
|April 26
| 243.333<br />(43.333)
|
| 3145728/2734375<br />(?)
| [[Magnesium]] (360d)
|-
|-
|120
| 20
|
| 149\360<br />(5\360)
|May 1
| 703.33<br />(43.33)
|
| 4/3<br />(126/125)
|-
| [[Degrees]]
|150
|
|June 1
|
|-
|180
|Symmetric tritone
|July 1
|
|-
|210
|Dodecaphonic perfect fifth
|August 1
|442/295
|-
|211
|Just perfect fifth
|August 2
|3/2
|-
|240
|
|September 1
|
|-
|270
|
|October 1
|
|-
|291
|Harmonic seventh
|October 21
|
|-
|300
|
|November 1
|
|-
|330
|
|December 1
|
|-
|360
|Octave
|January 1
|
|}
|}
==Music==
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
* [https://www.youtube.com/watch?v=VSKqwJkWu_U Idyllic Tribe] by [[User:Eliora|Eliora]]
 
== Music ==
; [[User:Eliora|Eliora]]
* [https://www.youtube.com/watch?v=VSKqwJkWu_U ''Idyllic Tribe''] (2022)
 
== Application as a logarithmic scale outside of music ==
360edo is used in the {{w|eyeborg}}, which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Sonifications]]
[[Category:Highly melodic]]
[[Category:Real-life sonifications]]
[[Category:Listen]]
[[Category:Listen]]
{{Todo| cleanup |comment=move trimisty away}}

Latest revision as of 13:31, 13 March 2026

← 359edo 360edo 361edo →
Prime factorization 23 × 32 × 5 (highly composite)
Step size 3.33333 ¢ 
Fifth 211\360 (703.333 ¢)
Semitones (A1:m2) 37:25 (123.3 ¢ : 83.33 ¢)
Dual sharp fifth 211\360 (703.333 ¢)
Dual flat fifth 210\360 (700 ¢) (→ 7\12)
Dual major 2nd 61\360 (203.333 ¢)
Consistency limit 7
Distinct consistency limit 7

360 equal divisions of the octave (abbreviated 360edo or 360ed2), also called 360-tone equal temperament (360tet) or 360 equal temperament (360et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 360 equal parts of about 3.33 ¢ each. Each step represents a frequency ratio of 21/360, or the 360th root of 2.

Theory

360edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. It can also be used with 2.5.9.13 subgroup.

In the 5-limit, the patent val supports the misty temperament, and in the 7-limit 360edo supports the trimisty (name proposed by Eliora) 63 & 99 temperament with the comma basis {10976/10935, 2097152/2083725}, which is similar to the misty temperament but has a period of 1/9- rather than 1/3-octave.

360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for degrees, the 140 & 220 temperament with period 1\20. Aside from that, it provides the optimal patent val for the 41 & 360 temperament with comma basis {10976/10935, 16384000000/16209796869}, on which it has lower badness than any other 7-limit temperament for which 360edo gives the optimal patent val. It also supports 12 & 360 with the comma basis {390625/388962, 67108864/66430125}.

Aside from the patent val, there is a number of mappings to be considered. The 360d val, ⟨360 571 836 1010], tempers out 3136/3125, 5120/5103, and extends the misty temperament in to the 7-limit. It is also a tuning for the 12th-octave magnesium temperament.

Odd harmonics

Approximation of odd harmonics in 360edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.38 +0.35 +1.17 -0.58 -1.32 -0.53 -1.60 -1.62 -0.85 -0.78 -1.61
Relative (%) +41.3 +10.6 +35.2 -17.3 -39.5 -15.8 -48.1 -48.7 -25.4 -23.4 -48.2
Steps
(reduced) 		571
(211) 		836
(116) 		1011
(291) 		1141
(61) 		1245
(165) 		1332
(252) 		1406
(326) 		1471
(31) 		1529
(89) 		1581
(141) 		1628
(188)

Subsets and supersets

360 is the 13th highly composite edo, with many proper divisors: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180. One step of 360edo is known as the Dröbisch angle, an interval size measure first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle".

Table of intervals

Eliora proposes notating 360edo with calendar dates, Jan 1 being the tonic, Jan 2 being the next step, etc, and each month having even 30 days. The notation is convenient because 1 month in this scenario is equal to 1 semitone, and corresponds to 12edo.

Any other notation system involving the number 360 can also be used.

See: Table of 360edo intervals

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 119\360 396.67 44/35 Squarschmidt
2 53\360 176.67 448/405 Quatracot
3 149\360
(29\360) 		703.33
(303.33) 		4/3
(135/128) 		Misty
4 23\360 76.67 4302592/4100625 Reenactment
9 149\360
(29\360) 		703.33
(36.67) 		4/3
(135/128) 		Trimisty
12 73\360
(13\360) 		243.333
(43.333) 		3145728/2734375
(?) 		Magnesium (360d)
20 149\360
(5\360) 		703.33
(43.33) 		4/3
(126/125) 		Degrees

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora

Application as a logarithmic scale outside of music

360edo is used in the eyeborg, which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.

Retrieved from "https://en.xen.wiki/index.php?title=360edo&oldid=225772"